Pierre Simon Laplace.

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very nearly equal to 30000". Therefore we shall suppose g = 30000", in [7188]
the equations [7170, 7171, 7173] ; and, after dividing them by h", we can

h h! h"'

deduce the values of — , — , — .
These must be substituted in the

tX it ili

equation [7172], after dividing it by h", and substituting ^ = 30000" in

the divisor ( 1 + ^^ ,, ) . Thus we shall obtain a more correct value of [7189]

g, which must be used, like the preceding value [7188], in repeating the „

operation ; and by continuing the calculation, in this way, we shall finally peculiar to

the third
obtain, satellite.

g^ == 29009",8 ; gg-f 154",63 = 29164",43 ; [7190]

. h = 0,02381 1 1 .h" ; log.coeff. = 8,37678 ; [7i9i]

h' = 0,2152920./i" ; log.coeff. = 9,33303 ; [7192]

h'" = —0,1291564.^" ; log.coeff. = 9,1 1212,. [7193]

These values of h, h', h!", being less than h", we may consider h" as being ^7^941

the peculiar excentricity of the third satellite, ivhose apsides have an annual

sideral motion of 29009", 8.

* (3565) We have seen, in [7170A], that this value of g ought to be increased to
17S483",9 nearly. JVIoreover the coefficient [7196] is printed 0,0020622 in the original ; t^^^^**!
we have corrected it for an error in the fifth decimal place. We may remark that the
comparison of the values of g, g^^, g^, g^ [7176, 7183,7190, 7195], corresponding to the '- ■•
peculiar motions of the perijoves of the first, second, third and fourth satelHtes respectively,
shows that they decrease rapidly with the mean distance of the satellite from Jupiter.
This peculiarity does not take place in the planetary orbits, as is observed in [7199'], and '• ^

as we may see by inspecting the first lines of the formulas [4242 — 4248]. The reason of
this difference is, that the motions of the perijoves of the satellites are produced chiefly by
the ellipticity of Jupiter's mass [7174'J.

[7198']

248 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Gel.

Lastly, the fourth value of g, or ^3, is that which is given by observation

f cuiiarto in [7124], for the annual sideral motion of the apsides of the fourth satellite ;

BStoK and we have found, in [7124, 7145' — 7147], that in this case we get,

[7195] ^3 = 7959",105 ; ^3+154",63 = 81 13",735 ;

[7196] h = 0,0020522f '[7 1 83a] ; log.coeff. = 7,31222 ;

[7197] h' = 0,0173350./i'" ; log.coeff. = 8,23892 ;

[7198] j^'^ 0,0816578.^'". log.coeff. = 8,91200.

These values of h, h\ h", being less than h'", we may consider h!" as the
peculiar excentricity of the fourth satellite, whose apsides have an annual

[7199] sideral motion of 7959",105.

Hence we see that each satellite has an excentricity ivhich is particularly

[719^] adapted to it. This peculiarity, which does not take place in the theory of
the planets [7183c], depends on the oblateness of Jupiter; the effects of this

[7200] oblateness on the perijoves of the satellites being very great. It now remains
to find the excentricities which are peculiar to each satellite, and the positions
of their apsides, at a given epoch. We shall give, in explaining the theory of
each satellite, what has been discovered by observation relative to this subject.

We shall now consider the inclinations and motions of the nodes of the orbits
of the satellites. These elements depend upon the equations in >• and /,

[7201] [6887 — 6894], which we shall here resume. The equations in >^
[6887 — 6890] become, by substituting the values of f*, m, m', m", m'"
[7141_7145],

[7202] 0=— 103",27+571269",64.x— . 9253",80.x'— 4606",32.V'— 327",25.x"' ;

[7203] =— 207",29— 5471",12.x+133377",33.x'— 173]5",94.x"— 769",65.x'" ;

[7204] = — 417",63— 566",l6.x— 3599",79.x'+28478",73.x"_2511",25.x'";

[7205] 0=— 974",19— 62",92.x_- 250",28.x'— 3928",28.x"+8179",ll.x'".

X x'

^], J„ Resolving these equations, we find,*

log.(l_x) =9,9997485;
log.(l— X') =9,9974480;
log.(l— X") =9,9880736;
log.(l—x'") = 9,9383406.

[7200']

X", X''

[7206] -^ =0,00057879; log.x =6,7625210

[7207] X' = 0,00585888 ; log.x' = 7,7678146

[7208] X" = 0,02708801 ; log.x" = 8,4327771

[7209] x'"= 0,13235804 ; log.x'"= 9,1217504

[7206a]

* (3566) For the purpose of verifying these calculations, the values of X', X", X"',

[7207 — 7209], were substituted in [7202], and the resulting value of X was found to be

0,00057 ; which differs from [7206] by an insensible quantity. In a similar way we have

[72066] jjeduced X' = 0,005858, from [7203] ; X"=0,027088, from [7204] ; and X'"=0,13230,

from [7209] ; which agree nearly with the results of the author in [7206 — 7209].

VIII. X. ^28J. ON THE INCLINATIONS OF THE ORBITS. 249

These values of K >^', '^", '^"\ determine the parts of the latitudes of the

... . . r7209'l

satellites, which depend upon the inclination of the equator of Jupiter to its

orbit. For we have, as in [6361], 200° — ^y' equal to the longitude of the t^^^^]
ascending node of the equator of Jupiter upon the orbit of this planet, and for

brevity we shall represent it by 1 = 200° — ^'; moreover, &' [6360] represents [7211]

the inclinations of these planes to each other. Then it follows, from [6362], I

that the parts of the latitudes of the satellites above Jupiter's orbit, [7212]
mentioned in [7209'], will be represented by the following expressions;*

s = (1— X). Lsm.{v — 1) ; [7213]

s'=={\ — >^'). ^'.sin.(«?' — I); [7214]

s" = (1— X"). ((. sin.(??"— I) ; [7215]

s"'={\—y!").^'.^m,{v"'—{). ' [7216]

The inclination &' of the equator of Jupiter to the orbit of this planet, and
the longitude I of the ascending node upon the same orbit, are determined
by observation. Delambre obtained, for the epoch of 1750,

a' = 3°,43519 ; log. ^' = 0,5359507 ; [7217]

I == 348^,62129. [7218]

These values of ^' and I are not rigorously constant. We have seen, in
[6928, 6929], that the value of &' increases annually by 0",07035, and that [7219]
the value of I decreases annually by 0",8259, relative to the fixed equinox.f
These quantities are so small that we need not take notice of them, during
the interval in which the eclipses of the satellites have been observed ; but it is
easy to introduce them in the calculation, if it be thought proper.

[7220]

* (3567) Substituting in the expression of the part of the latitude of the satellite m
[6362], the value of ^' = 200°— I [7211], it becomes,

(X— l).^'.sin.(t> + 200°— I) =(1— X).6'.sin.(u— I), [7212a]

as in [7213]; and by changing successively the quantities v, X, corresponding to the
satellite m, into v', X'; v", X" ; v'", X'", corresponding respectively to the other satellites, [72126]
we obtain the parts of the values of /, s", s'" [7214, 7215, 7216].

t (3568) The value of I [7218] being substituted in ^'=200°— I [7211], gives at
the epoch of 1750, *' = — 148^,62129 ; which is used in [6929i]. Moreover we have [^^^^«]
shown, in [6929^], that the general expression of I is I = 348°,62129— 0'',8259.<; so [72196]
that its annual decrement is 0",8259, as in [7219].

VOL. IV. 63

[7225']

250 MOTIONS OF THE SATELLITES OF JUPITER.^ [Mec. Cel.

The equations in / [6891 — 6894],* become, by substituting the values of
fi., m, m', m", m'" [7141,&c.],
[7221] = (;?— 571269",64)./+9253",80./'+4606",32i"+327",25./'" ; (5)

[7222] = 5471",12J+(j9— 133377",33)./'+17315",94./"+769",65.Z"' ; (6)

[7223] = 566",16J+3599",79.Z'+(;?— 28478",73)./"+251 i",25./"'; (7)

[7224] = 62",92./+250",28./'+3928",28.Z"+(;7— 8179",11)./'". (8)

These four equations give an equation in p, of the fourth degree. f To
obtain the roots we can use the approximate method, which is employed in
finding the values of g [7175, &c.]. In this manner we shall have the first
value of p, relative to the orbit of the first satellite, by putting the coefficient
[7225] of /, in the equation [7221], equal to zero; which gives p = 571269",64.
Substituting this in the equations [7222, 7223,7224], we may thence deduce

v I" r" , . .

the values of y , y , — . Then substituting these values in the equation

[7221], after dividing it by /, we shall obtain a more approximate value of
p. This value must then be used instead of the former, and the operation
must be repeated, till the two consecutive values of p shall differ but very
little from each other. By this means we shall obtain, after a few
operations, t

* (3569) The coefficients of the equations [6891 — 6894], are the same as those in

■* [6887 — 6890], which are computed in [7202 — 7204], and agree with the numerical values

in [7221 — 7224]. We may moreover observe, that the equations [6891 — 6894] can be

[722161 ^^^^^^^ ^^^^ [6887 — 6890], by changing the last terms of these equations, namely;

— 103",27; —207^29; —417^63; — 974",I9, into pi, pV, pi", pi'", respectively;

also X, X', X", X'", into — I, — /', — /", — /'", respectively. Now making the same

[7221c] changes in [7202 — 7205], which are derived from [6887 — 6890], we get, without any

reduction, the equations [7221—7224].

t (3570) Finding I from [7221], and substituting it in [7222—7224], we get three
'■ *'■' equations in l', I", I"'. Then finding /' from the first of these equations, and substituting

it in the others, we get two equations in I", V". From the first of these, we get V ', then
[72245] substituting it in the second, and dividing it by V", we get an equation of the fourth degree

in p.

X (3571) To verify the values [7226 — 7229], we have reduced the equations
[7226a] [7221 — 7224] by dividing them by I ; then using these new forms, we have substituted in

I' I" I'"
[7221] the expressions of y, y, —[7227 — 7229], and have obtained the value of

peculiar
to the first
satellite.

[7230]

VIlI.x.«^28.] ON THE INCLINATIONS OF THE ORBITS. 251

;, = 571389",32 ; ;?— 154",63 = 571234",69 ; [7226]

/' = —0,0124527. 1 ; log.coefl. = 8,09526„ ; [7227]

/" = —0,0009597. 1 ; log.coeff. = 6,98214„ ; [7228]

/'" = —0,0000995. /. log.coeff. = 5,99782,. [7229]

The values of V, I", /'", being in this case less than /, we may consider the ^'. '

quantity I as expressing the peculiar inclinations of the orbit of the first

satellite, upon a plane which passes always through the nodes of Jupiter'' s

equator, betiveen that equator and the orbit of the planet, and inclined to the

equator by the angle >^^'.* If we substitute the preceding values of >^, 6'

[7206, 7217], we shall find this inclination to be x^' = 19",88. The [7231]

preceding value of p [7226] then expresses the annual retrograde motion of

the nodes of the orbit upon this plane ;f consequently this motion is [7232]

571389",32 [7226].

I'
p [7226]. From the equation [7222], we get -r [7227], by the substitution of the values

I" I'" I"

of i?, T' T ['^226, 7228, 7229]. In like manner, from [7223] we find — [7228], by [72266]
It "

I' I"' I '"

the substitution of the values of p, y , y [7226, 7227, 7229]. Finally we obtain y

,, ,. [7226c]

[7229], from [7224], by the substitution of the values of jt?, y , y [7226—7228]. A

similar process was used in the verification of the other expressions [7233 — 7248] ;
observing that in the verification of the values [7233 — 7236], we have divided the equations [7226rf]
[7221 — 7224] by l'; in those of [7238 — 7241] we have divided the equations
[7221 — 7224] by /" ; and finally, in the verification of the values [7245—7248], we have
divided the equations [7221 — 7224] by l'". The results of these verifications confirm, to [7226e]
a sufficient degree of accuracy, the correctness of the numerical results, deduced by the
author from the proposed system of equations [7221 — 7224],

[7230a]

[72306]

* (3572) This agrees with what is said in [6357 — 6362] ; where it is shown that the
proposed fixed plane is inclined to the orbit of Jupiter by the angle (1 — X).^' ; subtracting
this from 6', the angle of inclination of Jupiter's orbit and equator [6360], we obtain X6f
for the inclination of this plane to the plane of Jupiter's equator, as in [7230] ; and by-
substituting the values of X, d' [7206, 7217], it becomes 19",88, as in [7231]. The
inclinations of the similar planes corresponding to the second, third and fourth satellites, are
X'd', X"6', X'"^', respectively, as in [7242, 7249, Stc] ; and by using the values [7230c]
[7208, 7209], we find that the two last of these quantities become as in [7243, 7250].

t (3573) We have seen, in [7133c], that pt expresses the retrograde motion of the
node of the satellite upon the fixed plane, and from the Jixed equinox [7328, &.C.] ; and its '■ •'

252 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Gel.

// The second value of p [7233] corresponds to the orbit of the second
[he"econd Satellite. It is given by observation [7133] ; and we have seen in the
satoihtc. p^(3ceding article [7133, 7148—7150], that we have in this case,
[7233] j9, = 133870",4 ; ^j^— 154",63 = 133715",77 ;

[7234] / =0,0207938./'; log.coeff. = 8,31793 ;

[7235] I" = —0,0342530. Z' ; log.coeff, = 8,53470, ;

[7236] r == —0,000931 2. /'. log.coeff. = 6,96904„.

The third value of p [7238] corresponds to the orbit of the third satellite;

we shall have the first approximation to its value, by putting the coefficient

of /" in the equation [7223] equal to nothing, which gives p = 28478",73.

[7237] Substituting this in [7221, 7222, 7224], we may thence deduce -the values

/ r v"

of — , — , — -. These values being substituted in [7223], after dividing

,„ it by /", give a second value of p ; which is to be used, in repeating the
peculiar to proccss, in a second operation ; and by proceeding in this manner, we finally
satellite, obtain,

[7238] p^ = 28375",48 ; p^—l 54",63 = 28220",85 ;

[7239] / = 0,01 1 1626. 1" ; log.coeff. = 8,04777 ;

[7240] /' = 0,1640530. /" ; log.coeff. = 9,21498 ;

[7241] /'"= — 0,1965650.Z" ; log.coeff. = 9,29351„.

These values of I, l', I'", are less than /", so that this quantity may be
considered as expressing the peculiar inclination of the orbit of the third
satellite upon a plaiie, ivhich passes always through the nodes of Jupiter^s
equator, between the equator and the orbit of the planet, and is inclined to the
equator by the quantity '^'^' [7230c]. Substituting for x" and ^ their values

[7243] [7208, 7217], we find this inclination to be 930",52. The annual retrograde
motion of the nodes of the orbit of the third satellite upon this plane, is
28375",48 [7238].

Lastly, the fourth value of p corresponds to the orbit of the fourth satellite.
We obtain a first approximation to its value, by putting the coefficient of /'"

[7244] i'l the equation [7224], equal to nothing, which gives ^=8179", 11.
Substituting this in [7221 — 7223], we get three equations for the

I r I"
determination of -— , — — . These values being substituted in the

£ t (/

mean value is the same as that on the plane mentioned in [7232] ; by subtracting the
[72316] precession 154",63.^, we get its value from the variable vernal equinox of the earth, so
that p — 154",63, represents this annual retrograde motion.

[7242]

[7243']

VIII. X. § 28.] ON THE INCLINATIONS OF THE ORBITS. 253

equation [7224], after dividing it bj l"\ give a second value of p^ which
must be used like the first value in repeating this process. In this manner
we finally obtain,

P3 = 7682",64 ; pg— 154",63 = 7528",01 ; [7245]

I =0,0019856./'
/' = 0,0234108./'
/"= 0,1248622./'

log.coeff. = 7,29789 ; [7246]

log.coeff. = 8,36942 ; [7247]
log.coeff. = 9,09643. [7248]

These values of /, /', /", are less than V" ; therefore /'" may he supposed p^, V",

to express the peculiar inclination of the fourth satellite, upon a plane ivhich Ihrfourth
passes always through the nodes of Jupiter^s equator, between the equator and
the orbit of the planet, and inclined to the equator by the angle x'"^' [7230c].
Substituting for x'" and &' their values [7209, 7217], we find this inclination [7249]

to be 4546",74. The annual retrosrrade motion of the nodes of the orbit , ^

' ^ ^ J [7250]

of the fourth satellite upon this plane, is 7682",64 [7245].

Hence loe see that the orbit of each satellite has an inclination ivhich is
peculiarly adapted to it ; a circumstance depending upon the oblateness
of Jupiter, whose influence upon the motions of the nodes of the orbits of the
satellites, is very great [7183c, (/]. It now remains to find the inclinations
corresponding to each orbit, and the positions of the nodes. We shall soon
see what has been discovered by observation, relative to this subject.

VOL. IV. 64

254 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Cel.

La Place's
first law.

[7252]

CHAPTER XL

ON THE LIBRATION OF THE THREE INNER SATELLITES OF JUPITER.

29. We have seen, in [6629], that the mean motions of the three inner

satellites of Jupiter are subjected to the following law, which holds good

relative to any variable axis, moving according to any law whatever [6632].

The mean motion of the first satellite^ plus twice that of the third, is

exactly equal to three times the mean motion of the second satellite.

To show how accurately this law agrees with observation, we shall give

the mean secular motions of these three bodies, as Delambre has determined

them, by the discussion of an immense number of eclipses. He has found

that, in one hundred Julian years, these motions, relative to the variable

Sons, equinox, are,*

[7253] First Satellite, 8258261°,63313;

[7254] Second Satellite, 4114125°,81277 ;

[7255] . Third Satellite, 2042057°,90398 ;

[7255'] ' [Fourth Satellite, 875427o,45956]. [7281]

[7253a]

* (3574) The motions of the satellites from the variable equinox in 100 Julian years,
is given in [7253 — 7255]. Dividing these by 100, we get the motions in one Julian year,
which is taken for the unit of time in [7283']. Subtracting the annual precession 154",63
[4357], we get the motions from the fixed equinox 825826009", 411412427",
204205635", 87542591", which agree nearly with the values of w, n', n", 7i"' [6025)t].
With the preceding value of n'% we obtain, from [6840], the expression of M = 337211"
[6025m] ; agreeing nearly with Bouvard's tables 337212",094. These values of
n, n', w", n'", agree very nearly with those used by Delambre, in his new tables
[7253c] [678U,&c.], as is evident by subtracting the precession for 100 years 1°,5463 from the
numbers in [6781o— r] ; then dividing by 100, and reducing to seconds.

VIII. xi. <^ 29.] LIBRATION OF THE SATELLITES. 255

The mean motion of the first, minus three times that of the second, plus
twice that of the third, is therefore equal to 27",8. This difference is so [7256]
small that it excites surprise, at the very near agreement of the theory with
the observations ; and as the tables must be strictly subjected to the
preceding law, the results in [7253 — 7255] have been slightly altered, by [7257]
Delambre, to attain this object.

We have seen, in [6630], that the epochs of the mean motions of the
three satellites are subjected to the following law : J;^

The epoch of the first satellite, minus three times that of the second, plus
twice that of the third, is exactly equal to the semi-circumference, or 200"^.

Delambre has determined these epochs, by the discussion of a very great
number of eclipses, and has obtained the following results, corresponding to

La Place's
sec on
]aw.

[7258]

t of the f

irst of Janua

ry, 1 750, at midnight ;

Epochs.

First

Satellite,

16°,69584;

[7259]

Second

Satellite,

346^,0521 ;

[7260]

Third

Satellite,

11°,41354;

[7261]

[Fourth

Satellite,

366^,89437 J. [7282]

[7261']

From these results of observation it appears, that the epoch of the first
satellite, minus three times that of the second, plus twice that of the third,
is equal to 200°,01962,* which exceeds the semi-circumference by 196",2. [7262]

* (3575) The numbers [7259—7261] give,

160,69584+3(400°— 346o,0521)-f2X 11°,4I354 = 201^,36662, [7259a]

instead of 200^,01962, which is given by the author in [7262]. This difference probably
arises from a mistake in the angle [7260], which is too small by about half a degree.
These angles are afterwards changed by the author, into 16^,68093, 346° ,48931, [72596]
11°,39349, in [7495,7439, 7386], respectively ; which satisfy the second law, or theorem,
on the epochs [7258], within 0°,00002. We may observe that these angles are not L'^^^c]
explicitly given in Delamhre^s tables ; for, instead of them, he uses the times of the mean
conjunction, or middle of the eclipse. From these times we find the epochs or angles [7259rf]
e, e', e", e'", corresponding to the commencement of the year 1750, to be nearly
16° ,67860, 346°,51677, 11°,39696, 366°,89437, respectively. The first three of [7259e]
these values agree nearly wiih those used by the author [72596] ; but the last, or the value
©"'=366°, 894.37, corresponding to the commencement of the year 1750, exceeds the
value 800,61249, given by the author in [7232, 7281«], by the quantity 286°,28188. [7259/]
This mistake of the author has been noticed by Professor Airy, in Vol. 6, page 98, of the
Memoirs of the Royal Astronomical Society of London. It has however no effect on the [72595"]
subsequent calculations of the coefficients of the inequalities, but may be considered as

256 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Gel.

Therefore the observations do not satisfy, quite so well, this second law

[7262'] relative to the epochs, as they do the first law, relative to the mean motions ;
and it would not have been strange if there had been found a still greater
difference, for the following reasons.

We are situated at so great a distance from the satellites of Jupiter, that
they disappear from our sight before they are wholly immersed in the

[7263] shadow of the planet ; and they do not again become visible until they have
partly emerged from it. To determine the time of the conjunction of a
satellite, we suppose, that, at the moment of the immersion, its centre is
at the same distance from the conical shadow as at the moment of the
emersion. Now it may happen, that the part of the disk of the satellite,
which first enters into the shadow, and of course re-appears the first, may
be more or less adapted to reflect the sun's light, than the part which is
eclipsed the last. In this case it is evident, that, at the moment of the
immersion, the distance of the centre of the satellite from the conical
shadow, will be greater or less than at the time of the emersion ; and the
time of conjunction, deduced from these observations, will therefore be more

[7264] or less advanced than the true time. The epochs of the mean longitudes
of the three inner satellites, deduced from the observations of their eclipses,
may differ, on this account, from their real values, and therefore may not
satisfy accurately the second law above mentioned [7258]. It is true that
we have here supposed that the part of the disk, which is first eclipsed, is,
in all cases, sensibly the same. Now this is really the case ; for it is well

^ known that the satellites present always the same face towards Jupiter, as

the moon does to the earth. The circumstance we have just mentioned
does not prevent the observations from satisfying the lavi^ of the mean motions
[7252], For these motions are determined by means of the difference of
the epochs at very distant intervals of time, and are therefore independent of

[7265] the inequalities, which might exist in the light of the different parts of the
disks of the satellites, particularly when we notice as many immersions as
emersions.

The difference between the result of the observations and the law of the

[7266] epochs [7258—7262], being very small, Delambre has thought it best to

[7259/1]

nothing more than an inaccurate deduction from the numbers in the tables ; and that the
mistake can be wholly corrected, by merely changing the angle into its corrected value,
wherever it occurs in the formulas.

I

VIII. xi. -^ 29.] . . LIBRATIOJV OF JHE SATELLITES. „ . 257

subject the epochs of his tables strictly to this law ; the corrections necessary r^o^p/,
to be made in the observations being within the limits of the errors to which
they are liable.

The tivo preceding theorems give rise, as ive have seen in [6657, &c.],
to a particular inequality, which we have denoted by the name of the r^n^-,

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